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Theorem distgp 23402
Description: Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
distgp.1 𝐵 = (Base‘𝐺)
distgp.2 𝐽 = (TopOpen‘𝐺)
Assertion
Ref Expression
distgp ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)

Proof of Theorem distgp
StepHypRef Expression
1 simpl 484 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ Grp)
2 simpr 486 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 = 𝒫 𝐵)
3 distgp.1 . . . . . 6 𝐵 = (Base‘𝐺)
43fvexi 6854 . . . . 5 𝐵 ∈ V
5 distopon 22299 . . . . 5 (𝐵 ∈ V → 𝒫 𝐵 ∈ (TopOn‘𝐵))
64, 5ax-mp 5 . . . 4 𝒫 𝐵 ∈ (TopOn‘𝐵)
72, 6eqeltrdi 2847 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐽 ∈ (TopOn‘𝐵))
8 distgp.2 . . . 4 𝐽 = (TopOpen‘𝐺)
93, 8istps 22235 . . 3 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝐵))
107, 9sylibr 233 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopSp)
11 eqid 2738 . . . . . 6 (-g𝐺) = (-g𝐺)
123, 11grpsubf 18785 . . . . 5 (𝐺 ∈ Grp → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1312adantr 482 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
144, 4xpex 7680 . . . . 5 (𝐵 × 𝐵) ∈ V
154, 14elmap 8768 . . . 4 ((-g𝐺) ∈ (𝐵m (𝐵 × 𝐵)) ↔ (-g𝐺):(𝐵 × 𝐵)⟶𝐵)
1613, 15sylibr 233 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺) ∈ (𝐵m (𝐵 × 𝐵)))
172, 2oveq12d 7370 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = (𝒫 𝐵 ×t 𝒫 𝐵))
18 txdis 22935 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐵 ∈ V) → (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵))
194, 4, 18mp2an 691 . . . . . 6 (𝒫 𝐵 ×t 𝒫 𝐵) = 𝒫 (𝐵 × 𝐵)
2017, 19eqtrdi 2794 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝐽 ×t 𝐽) = 𝒫 (𝐵 × 𝐵))
2120oveq1d 7367 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝒫 (𝐵 × 𝐵) Cn 𝐽))
22 cndis 22594 . . . . 5 (((𝐵 × 𝐵) ∈ V ∧ 𝐽 ∈ (TopOn‘𝐵)) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2314, 7, 22sylancr 588 . . . 4 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (𝒫 (𝐵 × 𝐵) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2421, 23eqtrd 2778 . . 3 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → ((𝐽 ×t 𝐽) Cn 𝐽) = (𝐵m (𝐵 × 𝐵)))
2516, 24eleqtrrd 2842 . 2 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽))
268, 11istgp2 23394 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopSp ∧ (-g𝐺) ∈ ((𝐽 ×t 𝐽) Cn 𝐽)))
271, 10, 25, 26syl3anbrc 1344 1 ((𝐺 ∈ Grp ∧ 𝐽 = 𝒫 𝐵) → 𝐺 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3444  𝒫 cpw 4559   × cxp 5630  wf 6490  cfv 6494  (class class class)co 7352  m cmap 8724  Basecbs 17043  TopOpenctopn 17263  Grpcgrp 18708  -gcsg 18710  TopOnctopon 22211  TopSpctps 22233   Cn ccn 22527   ×t ctx 22863  TopGrpctgp 23374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7308  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7914  df-2nd 7915  df-map 8726  df-0g 17283  df-topgen 17285  df-plusf 18456  df-mgm 18457  df-sgrp 18506  df-mnd 18517  df-grp 18711  df-minusg 18712  df-sbg 18713  df-top 22195  df-topon 22212  df-topsp 22234  df-bases 22248  df-cn 22530  df-cnp 22531  df-tx 22865  df-tmd 23375  df-tgp 23376
This theorem is referenced by: (None)
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